Hybrid eTFCE–GRF: Exact Cluster-Size Retrieval with Analytical p -Values for Voxel-Based Morphometry

Abstract Threshold-free cluster enhancement (TFCE) improves sensitivity in voxel-wise neuroimaging inference by integrating cluster extent across all thresholds, but its reliance on permutation testing makes it prohibitively slow for large datasets. Probabilistic TFCE (pTFCE) replaces permutations with analytical Gaussian random field (GRF) p -values, which reduces runtime by more than an order of magnitude, yet relies on a fixed threshold grid that introduces discretisation error. Exact TFCE (eTFCE) eliminates this discretisation by computing the integral exactly via a union-find data structure, but still requires permutations for inference. We propose a hybrid method that combines eTFCE’s union-find data structure for exact cluster-size retrieval with pTFCE’s analytical GRF inference. The union-find builds the full cluster hierarchy in a single pass over sorted voxels and enables exact cluster-size queries at any threshold in near-constant time; GRF theory then converts these sizes into analytical p -values without permutations. We validate the method through a six-experiment Monte Carlo study on synthetic phantoms (64 3 , 80 subjects): null family-wise error rate is controlled at the nominal level (0/200 rejections, 95% CI [0.0%, 1.9%]); power curves match baseline pTFCE (Dice ≥ 0.999 at sufficient signal); smoothness estimation error is below 1%; and cross-variant concordance exceeds r = 0.99. On real brain data from UK Biobank ( N = 500, within-vendor) and IXI ( N = 563, cross-vendor), the method detects biologically plausible scanner, age, and sex effects; on IXI, significance maps form strict subsets of the reference R pTFCE output, which supports conservative family-wise error control. Both methods are implemented in pytfce , a pure-Python package with no R or FSL dependencies, available on PyPI. The baseline reimplementation completes whole-brain voxel-based morphometry in ∼ 5 s (75 × faster than R pTFCE), while the hybrid variant completes in ∼ 85 s (4.6 × faster) with the advantage of exact cluster-size retrieval; both are more than three orders of magnitude faster than permutation-based TFCE. Highlights Hybrid eTFCE–GRF: exact cluster-size retrieval with analytical inference FWER controlled at nominal level in Monte Carlo validation pytfce package 4.6 × (hybrid) to 75 × (baseline) faster than R pTFCE Validated on UK Biobank ( N = 500) and IXI ( N = 563) Open-source Python package: pip install pytfce